Asymptotic distribution of eigenvalues of Schrödinger operators with nonclassical potentials

Eigenvalue Asymptotics of Schrödinger Operators with Only Discrete Spectrum

“…By Lemma 2.3 in [15], V(x, y) = \xy\* is an ^4^-weight on R 2 and, by Remark 2.3 in [15], it is easily proved that the Schrodinger operator T'=-A + \xy\* has only discrete spectrum. We shall prove…”

“…Our method is a modification of that of Tachizawa [15] and we can apply it to some nonclassical potentials whose zero sets are not cones. The potentials which we consider in this section belong to the special function class, that is, A ^-weights and we use some results about ^4^-weights in Tachizawa [15,Section 2].…”

“…A nonclassical potential is the potential whose zero set is an unbounded subset, for example, V(x, y) = \xy\* on R 2 . The nonclassical potentials are studied by Gurarie [8], Levendorskii [9], Robert [11], Simon [13], Solomyak [14], and Tachizawa [15], and they only consider potentials whose zero sets are cones. We will outline the content of this paper.…”

“…A nonclassical potential is the potential V such that F>0 and the set {xeR n : V(x) = 0} is an unbounded set. Several results on these nonclassical potentials are known ( [8], [9,Section 10] [11], [13], [14], [15]) and those are only on the potentials whose zero sets are cones in R". Our method is a modification of that of Tachizawa [15] and we can apply it to some nonclassical potentials whose zero sets are not cones.…”

“…Several results on these nonclassical potentials are known ( [8], [9,Section 10] [11], [13], [14], [15]) and those are only on the potentials whose zero sets are cones in R". Our method is a modification of that of Tachizawa [15] and we can apply it to some nonclassical potentials whose zero sets are not cones. The potentials which we consider in this section belong to the special function class, that is, A ^-weights and we use some results about ^4^-weights in Tachizawa [15,Section 2].…”